A334980 a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 3_2-Dyck paths of length 4*n. A 3_2-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -2.

0, 3, 31, 248, 1941, 15334, 122915, 999456, 8231740, 68562887, 576661761, 4891506968, 41801697070, 359574305580, 3111012673755, 27055673506128, 236387476114548, 2073957836402524, 18264689865840284, 161403223665821280, 1430768729986730685, 12719497076318052990

Offset

0,2

Comments

For n = 1, there is no (n-1)-th up step, a(1) = 3 is the total number of down steps before the first up step.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1027

Andrei Asinowski, Benjamin Hackl, and Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020-2022.

Formula

a(0) = 0 and a(n) = 3*binomial(4*n+7, n+1)/(4*n+7) - 12*binomial(4*n+3, n)/(4*n+3) for n > 0.

a(n) ~ 37 * 2^(8*n+13/2) / (3^(3*n+11/2) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 23 2025

Example

For n = 2, the 3_2-Dyck paths are UDDDDDUD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD, DUDDDDUD, DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, DDUDDDUD, DDUDDUDD, DDUDUDDD, DDUUDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 5 + 4 + 3 + 2 + 1 + 0 + 4 + 3 + 2 + 1 + 0 + 3 + 2 + 1 + 0 = 31.

Mathematica

a[0] = 0; a[n_] := 3*Binomial[4*n+7, n+1]/(4*n + 7) - 12 * Binomial[4*n + 3, n]/(4*n + 3); Array[a, 22, 0]

Prog

(SageMath) [3*binomial(4*n + 7, n + 1)/(4*n + 7) - 12*binomial(4*n + 3, n)/(4*n + 3) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020

Crossrefs

Cf. A334976, A334977, A334978, A334979.

Sequence in context: A114654 A198151 A197231 * A111400 A057972 A221821

Adjacent sequences: A334977 A334978 A334979 * A334981 A334982 A334983

Keyword

nonn,easy

Author

Sarah Selkirk, May 18 2020

Status

approved